Thermal Conductivity Spectroscopy Technique to Measure Phonon Mean Free Paths
Open accessAbstract
Size effects in heat conduction, which occur when phonon mean free paths (MFPs) are comparable to characteristic lengths, are being extensively explored in many nanoscale systems for energy applications. Knowledge of MFPs is essential to understanding size effects, yet MFPs are largely unknown for most materials. Here, we introduce the first experimental technique which can measure MFP distributions over a wide range of length scales and materials. Using this technique, we measure the MFP distribution of silicon for the first time and obtain good agreement with first-principles calculations.
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The calculation overpredicts the effective thermal conductivity for all pump diameters below T=150 K.
“our calculation overpredicts the effective thermal conductivity for all pump diameters below T ¼ 150 K”
From Radial quasiballistic transport in time-domain thermoreflectance studied using Monte Carlo simulations (2014) · cites this paper as [14]
The main text (p. 2) reports that measured thermal conductivity at 90 K was below the accepted value for two pump diameters, and that measurements diverged from literature values below about 200 K. It also compares a calculated conductivity with measurements in Fig. 2(a), but describes their agreement as reasonably good; the text does not establish that the calculation overpredicts measurements for every pump diameter below 150 K. (LLM, conf=0.76)
Experimental — accuracy not yet measuredThe majority of experimental work in the literature deduced thermal properties based on Fourier's law of heat conduction with an effective thermal conductivity.
“Similarly, while TDTR can be used to study non-equilibrium phenomena such as quasi-ballistic phonon transport 90, 91, 109-111 and electronphonon coupling 55 due to its ability to detect temperature evolutions at micrometer-scale and picosecond-scale resolutions, the majority of the experimental work in the literature deduced thermal properties based on Fourier's”
From Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials (2018) · cites this paper as [91]
The paper’s main text (page not specified in our copy) says that transient thermoreflectance extracts thermal properties by fitting temperature-decay data to a thermal model, and it reports fitted thermal conductivities that differ with heater size. It does not establish that a majority of experimental work in the literature used Fourier’s law with an effective thermal conductivity. (LLM, conf=0.82)
Experimental — accuracy not yet measuredAt 80 K, phonon mean free paths are approximately an order of magnitude longer than at room temperature.
“Secondly, the phonon MFPs at 80 K are approximately an order of magnitude longer than those at RT [14, 19, 28].”
From Boundary conditions dictate frequency dependence of thermal conductivity in silicon (2025) · cites this paper as [14]
The paper’s main text (p. 095901-2) says phonon MFPs are shorter at room temperature and increase rapidly as temperature decreases, supporting the direction of the comparison. It does not give an 80 K-to-room-temperature MFP ratio or establish the claimed order-of-magnitude difference. (LLM, conf=0.84)
Experimental — accuracy not yet measuredAt 80 K, the Al/Si measurements align closely with first-principles results interpolated from Minnich et al. at 100 K and 60 K, with a slight deviation for phonons with short mean free paths.
“At 80 K, our measurements align closely with first-principles results, calculated through interpolation of calculations by Minnich et al. at 100 K and 60 K [14], with a slight deviation for phonons with short MFPs.”
From Boundary conditions dictate frequency dependence of thermal conductivity in silicon (2025) · cites this paper as [14]
The main text and Fig. 3 caption (p. 3) identify Minnich et al.’s first-principles calculations of silicon thermal conductivity versus phonon mean free path, so the citation supports the source of the calculations. Our copy does not identify calculations specifically at 100 K and 60 K. The 80 K interpolation, agreement with the citing authors’ measurements, and short-MFP deviation are presented as the citing authors’ own analysis, not findings attributed to Minnich et al. (LLM, conf=0.78)
Experimental — accuracy not yet measuredThe ballistic thermal resistance effects found in TDTR measurements in both this and previous works can contribute over a wide temperature range extending from cryogenic up to even room temperatures.
“These findings are consistent with the conclusion that the ballistic thermal resistance effects found in TDTR measurements in both this and previous [39] works can contribute over a wide temperature range extending from cryogenic up to even room temperatures.”
From Mean Free Path Effects on the Experimentally Measured Thermal Conductivity of Single-Crystal Silicon Microbridges (2013) · cites this paper as [39]
The paper’s TTR measurements section (page not specified in our copy) reports ballistic thermal resistance in silicon as temperature falls below about 200 K. It also says the measurements are independent of beam diameter and agree with literature values at room temperature. Thus it supports the cryogenic effect, but not the claim that the effects found in this work extend to room temperature. (LLM, conf=0.89)
Experimental — accuracy not yet measured
This interpretation is the same as that of Koh and Cahill and consistent with Minnich et al. and Johnson et al., who instead used the laser spot diameter and transient grating period as the cutoff dimensions.
“This interpretation is the same as that of Koh and Cahill 16 , and consistent with Minnich et al. 1 and Johnson et al. 19,20 who instead used the laser spot diameter and transient grating period as the cutoff dimensions.”
From Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance (2013) · cites this paper as [1]
Our data compare favourably with the first principles predictions by Esfarjani and coworkers at all temperatures.
“Although our data compare favourably with the first principles predictions by Esfarjani and coworkers 1,2,41 at all temperatures”
From Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance (2013) · cites this paper as [1]
The pump size is a key variable that sets the thermal length scale for radial transport.
“The diameter dependence of the radial heat flux demonstrates that the pump size is a key variable that sets the thermal length scale for radial transport, confirming previous explanations for observations of a pump-beam size dependent thermal conductivity.14”
From Radial quasiballistic transport in time-domain thermoreflectance studied using Monte Carlo simulations (2014) · cites this paper as [14]
The effective thermal conductivity decreases with decreasing pump beam size.
“The results show the experimentally observed trend of decreasing effective thermal conductivity with decreasing pump beam size.14”
From Radial quasiballistic transport in time-domain thermoreflectance studied using Monte Carlo simulations (2014) · cites this paper as [14]
At cryogenic temperatures, the calculation predicts a similar trend in thermal conductivity versus pump size as experimental measurements.
“At these temperatures, our result predicts a similar trend in thermal conductivity versus pump size”
From Radial quasiballistic transport in time-domain thermoreflectance studied using Monte Carlo simulations (2014) · cites this paper as [14]
It is possible to experimentally measure the MFP dependent contributions of phonons to thermal conductivity (the phonon MFP spectrum).
“Recent studies have shown that it is possible to experimentally measure the MFP dependent contributions of phonons to thermal conductivity–the phonon MFP spectrum12345.”
From Universal phonon mean free path spectra in crystalline semiconductors at high temperature (2013) · cites this paper as [3]
Further studies using TDTR have equated l* to the laser spot size on Si and the dimensions of nano-patterned heaters on sapphire.
“Further studies using TDTR have equated l* to the laser spot size on Si3 and the dimensions of nano-patterned heaters on sapphire26.”
From Universal phonon mean free path spectra in crystalline semiconductors at high temperature (2013) · cites this paper as [3]
Heat sources with dimensions smaller than phonon MFPs perceive a locally suppressed thermal conductivity.
“Heat sources with dimensions smaller than phonon MFPs also perceive a locally suppressed thermal conductivity323.”
From Universal phonon mean free path spectra in crystalline semiconductors at high temperature (2013) · cites this paper as [3]
Care should be taken when applying a diffusive thermal transport model to systems where a significant fraction of heat is carried by phonons with long mean-free paths (MFPs).
“However, care should be taken when applying a diffusive thermal transport model to systems where a significant fraction of heat is carried by phonons with long mean-free paths (MFPs). 90-92, 112”
From Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials (2018) · cites this paper as [91]
The thermal conductivity reductions reported by Minnich et al. for comparable pump diameters in natural silicon agree in trend over the reported temperature range with the calculated reduction.
“Furthermore, the thermal conductivity reductions reported by Minnich et al. [39] for comparable pump diameters in natural silicon agree in trend over the reported temperature range, as shown in Fig. 7.”
From Mean Free Path Effects on the Experimentally Measured Thermal Conductivity of Single-Crystal Silicon Microbridges (2013) · cites this paper as [39]
Minnich et al.'s measurements were independent of modulation frequency but showed a pump diameter dependence.
“This is consistent with the results of Minnich et al. whose measurements were independent of modulation frequency (i.e., varying thermal penetration depth), but instead showed a pump diameter dependence.”
From Mean Free Path Effects on the Experimentally Measured Thermal Conductivity of Single-Crystal Silicon Microbridges (2013) · cites this paper as [39]
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- Authors
- Minnich, A. J., Johnson, J. A., Schmidt, A. J., Esfarjani, K., Dresselhaus, M. S., Nelson, K. A., & Chen, G.
- Affiliations
- Massachusetts Institute of Technology (US)
- Boston University (US)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2011-08-25
- Volume
- 107
- Issue
- 9
- Pages
- 095901-095901
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2019490594
- Cited by
- 556
- References
- 21
- Citations by year
- 2026: 24 · 2025: 42 · 2024: 25 · 2023: 23 · 2022: 33 · 2021: 48 · 2020: 38 · 2019: 40 · 2018: 35 · 2017: 34 · 2016: 66 · 2015: 54 · 2014: 34 · 2013: 34 · 2012: 20
- FWCI
- 14.77
- Type
- journal article
- Language
- en
- Open access
- green
- www.osti.gov
- Licence
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) (submittedVersion)
- DSpace@MIT (Massachusetts Institute of Technology) (submittedVersion, cc-by-nc)
- APS
- Funders
- U.S. Department of Energy
- National Science Foundation
- Topics
- Physical Sciences › Materials Science › Materials Chemistry › Thermal properties of materials
- Physical Sciences › Engineering › Mechanics of Materials › Thermoelastic and Magnetoelastic Phenomena
- Physical Sciences › Engineering › Civil and Structural Engineering › Thermal Radiation and Cooling Technologies
- Keywords
- phonon mean free path, phonon transport, silicon, nanoscale heat conduction, thermal conductivity measurement
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